LMFDB - Embedded newform 4410.2.d.c.4409.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(4409,4410)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4410.4409");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4409.12
Character	$\chi$	$=$	4410.4409
Dual form			4410.2.d.c.4409.11

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +(-0.526370 + 2.17323i) q^{5} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +(-0.526370 + 2.17323i) q^{5} -1.00000 q^{8} +(0.526370 - 2.17323i) q^{10} -4.41847i q^{11} -4.94214 q^{13} +1.00000 q^{16} +5.22951i q^{17} +4.97179i q^{19} +(-0.526370 + 2.17323i) q^{20} +4.41847i q^{22} +7.71767 q^{23} +(-4.44587 - 2.28785i) q^{25} +4.94214 q^{26} -10.2237i q^{29} +6.43420i q^{31} -1.00000 q^{32} -5.22951i q^{34} +9.25293i q^{37} -4.97179i q^{38} +(0.526370 - 2.17323i) q^{40} +6.79316 q^{41} +2.08223i q^{43} -4.41847i q^{44} -7.71767 q^{46} -7.83497i q^{47} +(4.44587 + 2.28785i) q^{50} -4.94214 q^{52} -5.67297 q^{53} +(9.60237 + 2.32575i) q^{55} +10.2237i q^{58} -12.1340 q^{59} -6.76571i q^{61} -6.43420i q^{62} +1.00000 q^{64} +(2.60139 - 10.7404i) q^{65} -2.40838i q^{67} +5.22951i q^{68} +10.4298i q^{71} -0.202603 q^{73} -9.25293i q^{74} +4.97179i q^{76} -0.736672 q^{79} +(-0.526370 + 2.17323i) q^{80} -6.79316 q^{82} +3.96971i q^{83} +(-11.3649 - 2.75266i) q^{85} -2.08223i q^{86} +4.41847i q^{88} -6.61837 q^{89} +7.71767 q^{92} +7.83497i q^{94} +(-10.8049 - 2.61700i) q^{95} -14.5487 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8}+O(q^{10}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8	$ 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{16} + 32 q^{23} - 16 q^{25} - 24 q^{32} - 32 q^{46} + 16 q^{50} - 32 q^{53} + 24 q^{64} - 32 q^{85} + 32 q^{92} + 32 q^{95}+O(q^{100}) $ 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^16 + 32 * q^23 - 16 * q^25 - 24 * q^32 - 32 * q^46 + 16 * q^50 - 32 * q^53 + 24 * q^64 - 32 * q^85 + 32 * q^92 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4410\mathbb{Z}\right)^\times$.

$n$	$1081$	$2647$	$3431$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−1.00000			−0.707107
$2$	−1.00000			−0.707107
$3$		0			0
$3$		0			0
$4$	1.00000			0.500000
$5$	−0.526370	+	2.17323i	−0.235400	+	0.971899i
$5$	−0.526370	+	2.17323i	−0.235400	+	0.971899i
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−1.00000			−0.353553
$9$		0			0
$10$	0.526370	−	2.17323i	0.166453	−	0.687236i
$11$		−	4.41847i		−	1.33222i	−0.745853	−	0.666110i	$-0.767958\pi$
$11$		−	4.41847i		−	1.33222i	0.745853	−	0.666110i	$-0.232042\pi$
$12$		0			0
$13$	−4.94214			−1.37070			−0.685351	−	0.728212i	$-0.740351\pi$
$13$	−4.94214			−1.37070			−0.685351	+	0.728212i	$0.740351\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$			5.22951i			1.26834i	0.773192	+	0.634172i	$0.218659\pi$
$17$			5.22951i			1.26834i	−0.773192	+	0.634172i	$0.781341\pi$
$18$		0			0
$19$			4.97179i			1.14061i	0.821434	+	0.570304i	$0.193174\pi$
$19$			4.97179i			1.14061i	−0.821434	+	0.570304i	$0.806826\pi$
$20$	−0.526370	+	2.17323i	−0.117700	+	0.485949i
$21$		0			0
$22$			4.41847i			0.942022i
$23$	7.71767			1.60925			0.804623	−	0.593786i	$-0.202368\pi$
$23$	7.71767			1.60925			0.804623	+	0.593786i	$0.202368\pi$
$24$		0			0
$25$	−4.44587	−	2.28785i	−0.889174	−	0.457570i
$26$	4.94214			0.969233
$27$		0			0
$28$		0			0
$29$		−	10.2237i		−	1.89850i	−0.314527	−	0.949248i	$-0.601846\pi$
$29$		−	10.2237i		−	1.89850i	0.314527	−	0.949248i	$-0.398154\pi$
$30$		0			0
$31$			6.43420i			1.15562i	0.816173	+	0.577808i	$0.196092\pi$
$31$			6.43420i			1.15562i	−0.816173	+	0.577808i	$0.803908\pi$
$32$	−1.00000			−0.176777
$33$		0			0
$34$		−	5.22951i		−	0.896854i
$35$		0			0
$36$		0			0
$37$			9.25293i			1.52117i	0.649237	+	0.760586i	$0.275088\pi$
$37$			9.25293i			1.52117i	−0.649237	+	0.760586i	$0.724912\pi$
$38$		−	4.97179i		−	0.806531i
$39$		0			0
$40$	0.526370	−	2.17323i	0.0832264	−	0.343618i
$41$	6.79316			1.06091			0.530457	−	0.847712i	$-0.322020\pi$
$41$	6.79316			1.06091			0.530457	+	0.847712i	$0.322020\pi$
$42$		0			0
$43$			2.08223i			0.317538i	0.987316	+	0.158769i	$0.0507524\pi$
$43$			2.08223i			0.317538i	−0.987316	+	0.158769i	$0.949248\pi$
$44$		−	4.41847i		−	0.666110i
$45$		0			0
$46$	−7.71767			−1.13791
$47$		−	7.83497i		−	1.14285i	−0.820655	−	0.571424i	$-0.806391\pi$
$47$		−	7.83497i		−	1.14285i	0.820655	−	0.571424i	$-0.193609\pi$
$48$		0			0
$49$		0			0
$50$	4.44587	+	2.28785i	0.628741	+	0.323551i
$51$		0			0
$52$	−4.94214			−0.685351
$53$	−5.67297			−0.779242			−0.389621	−	0.920975i	$-0.627394\pi$
$53$	−5.67297			−0.779242			−0.389621	+	0.920975i	$0.627394\pi$
$54$		0			0
$55$	9.60237	+	2.32575i	1.29478	+	0.313604i
$56$		0			0
$57$		0			0
$58$			10.2237i			1.34244i
$59$	−12.1340			−1.57971			−0.789855	−	0.613294i	$-0.789844\pi$
$59$	−12.1340			−1.57971			−0.789855	+	0.613294i	$0.789844\pi$
$60$		0			0
$61$		−	6.76571i		−	0.866260i	−0.901331	−	0.433130i	$-0.857409\pi$
$61$		−	6.76571i		−	0.866260i	0.901331	−	0.433130i	$-0.142591\pi$
$62$		−	6.43420i		−	0.817144i
$63$		0			0
$64$	1.00000			0.125000
$65$	2.60139	−	10.7404i	0.322663	−	1.33218i
$66$		0			0
$67$		−	2.40838i		−	0.294231i	−0.989119	−	0.147115i	$-0.953001\pi$
$67$		−	2.40838i		−	0.294231i	0.989119	−	0.147115i	$-0.0469989\pi$
$68$			5.22951i			0.634172i
$69$		0			0
$70$		0			0
$71$			10.4298i			1.23779i	0.785474	+	0.618895i	$0.212419\pi$
$71$			10.4298i			1.23779i	−0.785474	+	0.618895i	$0.787581\pi$
$72$		0			0
$73$	−0.202603			−0.0237129			−0.0118564	−	0.999930i	$-0.503774\pi$
$73$	−0.202603			−0.0237129			−0.0118564	+	0.999930i	$0.503774\pi$
$74$		−	9.25293i		−	1.07563i
$75$		0			0
$76$			4.97179i			0.570304i
$77$		0			0
$78$		0			0
$79$	−0.736672			−0.0828821			−0.0414411	−	0.999141i	$-0.513195\pi$
$79$	−0.736672			−0.0828821			−0.0414411	+	0.999141i	$0.513195\pi$
$80$	−0.526370	+	2.17323i	−0.0588500	+	0.242975i
$81$		0			0
$82$	−6.79316			−0.750179
$83$			3.96971i			0.435732i	0.975979	+	0.217866i	$0.0699096\pi$
$83$			3.96971i			0.435732i	−0.975979	+	0.217866i	$0.930090\pi$
$84$		0			0
$85$	−11.3649	−	2.75266i	−1.23270	−	0.298568i
$86$		−	2.08223i		−	0.224533i
$87$		0			0
$88$			4.41847i			0.471011i
$89$	−6.61837			−0.701546			−0.350773	−	0.936461i	$-0.614081\pi$
$89$	−6.61837			−0.701546			−0.350773	+	0.936461i	$0.614081\pi$
$90$		0			0
$91$		0			0
$92$	7.71767			0.804623
$93$		0			0
$94$			7.83497i			0.808115i
$95$	−10.8049	−	2.61700i	−1.10855	−	0.268499i
$96$		0			0
$97$	−14.5487			−1.47719			−0.738597	−	0.674147i	$-0.764511\pi$
$97$	−14.5487			−1.47719			−0.738597	+	0.674147i	$0.764511\pi$
$98$		0			0
$99$		0			0
$100$	−4.44587	−	2.28785i	−0.444587	−	0.228785i
$101$	−5.16164			−0.513602			−0.256801	−	0.966464i	$-0.582668\pi$
$101$	−5.16164			−0.513602			−0.256801	+	0.966464i	$0.582668\pi$
$102$		0			0
$103$	4.62070			0.455292			0.227646	−	0.973744i	$-0.426897\pi$
$103$	4.62070			0.455292			0.227646	+	0.973744i	$0.426897\pi$
$104$	4.94214			0.484617
$105$		0			0
$106$	5.67297			0.551008
$107$	−10.1677			−0.982944			−0.491472	−	0.870893i	$-0.663541\pi$
$107$	−10.1677			−0.982944			−0.491472	+	0.870893i	$0.663541\pi$
$108$		0			0
$109$	−9.18677			−0.879933			−0.439966	−	0.898014i	$-0.645010\pi$
$109$	−9.18677			−0.879933			−0.439966	+	0.898014i	$0.645010\pi$
$110$	−9.60237	−	2.32575i	−0.915550	−	0.221752i
$111$		0			0
$112$		0			0
$113$	7.19926			0.677249			0.338625	−	0.940922i	$-0.390038\pi$
$113$	7.19926			0.677249			0.338625	+	0.940922i	$0.390038\pi$
$114$		0			0
$115$	−4.06235	+	16.7723i	−0.378816	+	1.56402i
$116$		−	10.2237i		−	0.949248i
$117$		0			0
$118$	12.1340			1.11702
$119$		0			0
$120$		0			0
$121$	−8.52291			−0.774810
$122$			6.76571i			0.612538i
$123$		0			0
$124$			6.43420i			0.577808i
$125$	7.31220	−	8.45765i	0.654023	−	0.756475i
$126$		0			0
$127$			4.64476i			0.412155i	0.978536	+	0.206078i	$0.0660700\pi$
$127$			4.64476i			0.412155i	−0.978536	+	0.206078i	$0.933930\pi$
$128$	−1.00000			−0.0883883
$129$		0			0
$130$	−2.60139	+	10.7404i	−0.228157	+	0.941996i
$131$	−6.16011			−0.538211			−0.269106	−	0.963111i	$-0.586728\pi$
$131$	−6.16011			−0.538211			−0.269106	+	0.963111i	$0.586728\pi$
$132$		0			0
$133$		0			0
$134$			2.40838i			0.208053i
$135$		0			0
$136$		−	5.22951i		−	0.448427i
$137$	11.6165			0.992462			0.496231	−	0.868190i	$-0.334717\pi$
$137$	11.6165			0.992462			0.496231	+	0.868190i	$0.334717\pi$
$138$		0			0
$139$		−	15.1955i		−	1.28886i	−0.764663	−	0.644431i	$-0.777094\pi$
$139$		−	15.1955i		−	1.28886i	0.764663	−	0.644431i	$-0.222906\pi$
$140$		0			0
$141$		0			0
$142$		−	10.4298i		−	0.875249i
$143$			21.8367i			1.82608i
$144$		0			0
$145$	22.2185	+	5.38146i	1.84515	+	0.446906i
$146$	0.202603			0.0167675
$147$		0			0
$148$			9.25293i			0.760586i
$149$		−	4.18853i		−	0.343138i	−0.985172	−	0.171569i	$-0.945116\pi$
$149$		−	4.18853i		−	0.343138i	0.985172	−	0.171569i	$-0.0548837\pi$
$150$		0			0
$151$	−16.3287			−1.32881			−0.664407	−	0.747371i	$-0.731316\pi$
$151$	−16.3287			−1.32881			−0.664407	+	0.747371i	$0.731316\pi$
$152$		−	4.97179i		−	0.403266i
$153$		0			0
$154$		0			0
$155$	−13.9830	−	3.38677i	−1.12314	−	0.272032i
$156$		0			0
$157$	−11.1873			−0.892845			−0.446423	−	0.894822i	$-0.647302\pi$
$157$	−11.1873			−0.892845			−0.446423	+	0.894822i	$0.647302\pi$
$158$	0.736672			0.0586065
$159$		0			0
$160$	0.526370	−	2.17323i	0.0416132	−	0.171809i
$161$		0			0
$162$		0			0
$163$			9.35228i			0.732527i	0.930511	+	0.366263i	$0.119363\pi$
$163$			9.35228i			0.732527i	−0.930511	+	0.366263i	$0.880637\pi$
$164$	6.79316			0.530457
$165$		0			0
$166$		−	3.96971i		−	0.308109i
$167$			2.72339i			0.210742i	0.994433	+	0.105371i	$0.0336030\pi$
$167$			2.72339i			0.210742i	−0.994433	+	0.105371i	$0.966397\pi$
$168$		0			0
$169$	11.4247			0.878826
$170$	11.3649	+	2.75266i	0.871651	+	0.211119i
$171$		0			0
$172$			2.08223i			0.158769i
$173$		−	3.90918i		−	0.297210i	−0.988897	−	0.148605i	$-0.952522\pi$
$173$		−	3.90918i		−	0.297210i	0.988897	−	0.148605i	$-0.0474782\pi$
$174$		0			0
$175$		0			0
$176$		−	4.41847i		−	0.333055i
$177$		0			0
$178$	6.61837			0.496068
$179$		−	24.3957i		−	1.82342i	−0.410832	−	0.911711i	$-0.634762\pi$
$179$		−	24.3957i		−	1.82342i	0.410832	−	0.911711i	$-0.365238\pi$
$180$		0			0
$181$		−	22.0352i		−	1.63786i	−0.573893	−	0.818930i	$-0.694568\pi$
$181$		−	22.0352i		−	1.63786i	0.573893	−	0.818930i	$-0.305432\pi$
$182$		0			0
$183$		0			0
$184$	−7.71767			−0.568954
$185$	−20.1088	−	4.87046i	−1.47842	−	0.358084i
$186$		0			0
$187$	23.1065			1.68971
$188$		−	7.83497i		−	0.571424i
$189$		0			0
$190$	10.8049	+	2.61700i	0.783866	+	0.189857i
$191$		−	26.8532i		−	1.94303i	−0.236983	−	0.971514i	$-0.576158\pi$
$191$		−	26.8532i		−	1.94303i	0.236983	−	0.971514i	$-0.423842\pi$
$192$		0			0
$193$		−	13.9798i		−	1.00629i	−0.864202	−	0.503145i	$-0.832176\pi$
$193$		−	13.9798i		−	1.00629i	0.864202	−	0.503145i	$-0.167824\pi$
$194$	14.5487			1.04453
$195$		0			0
$196$		0			0
$197$	−11.2214			−0.799493			−0.399746	−	0.916626i	$-0.630902\pi$
$197$	−11.2214			−0.799493			−0.399746	+	0.916626i	$0.630902\pi$
$198$		0			0
$199$			1.80074i			0.127651i	0.997961	+	0.0638254i	$0.0203301\pi$
$199$			1.80074i			0.127651i	−0.997961	+	0.0638254i	$0.979670\pi$
$200$	4.44587	+	2.28785i	0.314370	+	0.161775i
$201$		0			0
$202$	5.16164			0.363171
$203$		0			0
$204$		0			0
$205$	−3.57572	+	14.7631i	−0.249739	+	1.03110i
$206$	−4.62070			−0.321940
$207$		0			0
$208$	−4.94214			−0.342676
$209$	21.9677			1.51954
$210$		0			0
$211$	1.70144			0.117132			0.0585659	−	0.998284i	$-0.481347\pi$
$211$	1.70144			0.117132			0.0585659	+	0.998284i	$0.481347\pi$
$212$	−5.67297			−0.389621
$213$		0			0
$214$	10.1677			0.695046
$215$	−4.52517	−	1.09603i	−0.308614	−	0.0747483i
$216$		0			0
$217$		0			0
$218$	9.18677			0.622207
$219$		0			0
$220$	9.60237	+	2.32575i	0.647391	+	0.156802i
$221$		−	25.8450i		−	1.73852i
$222$		0			0
$223$	3.22392			0.215890			0.107945	−	0.994157i	$-0.465573\pi$
$223$	3.22392			0.215890			0.107945	+	0.994157i	$0.465573\pi$
$224$		0			0
$225$		0			0
$226$	−7.19926			−0.478888
$227$		−	2.34174i		−	0.155427i	−0.996976	−	0.0777135i	$-0.975238\pi$
$227$		−	2.34174i		−	0.155427i	0.996976	−	0.0777135i	$-0.0247620\pi$
$228$		0			0
$229$		−	11.6386i		−	0.769102i	−0.923104	−	0.384551i	$-0.874356\pi$
$229$		−	11.6386i		−	0.769102i	0.923104	−	0.384551i	$-0.125644\pi$
$230$	4.06235	−	16.7723i	0.267863	−	1.10593i
$231$		0			0
$232$			10.2237i			0.671220i
$233$	−0.0869816			−0.00569836			−0.00284918	−	0.999996i	$-0.500907\pi$
$233$	−0.0869816			−0.00569836			−0.00284918	+	0.999996i	$0.500907\pi$
$234$		0			0
$235$	17.0272	+	4.12409i	1.11073	+	0.269026i
$236$	−12.1340			−0.789855
$237$		0			0
$238$		0			0
$239$			13.6099i			0.880351i	0.897912	+	0.440175i	$0.145084\pi$
$239$			13.6099i			0.880351i	−0.897912	+	0.440175i	$0.854916\pi$
$240$		0			0
$241$		−	5.13807i		−	0.330972i	−0.986212	−	0.165486i	$-0.947081\pi$
$241$		−	5.13807i		−	0.330972i	0.986212	−	0.165486i	$-0.0529193\pi$
$242$	8.52291			0.547874
$243$		0			0
$244$		−	6.76571i		−	0.433130i
$245$		0			0
$246$		0			0
$247$		−	24.5713i		−	1.56343i
$248$		−	6.43420i		−	0.408572i
$249$		0			0
$250$	−7.31220	+	8.45765i	−0.462464	+	0.534909i
$251$	11.2861			0.712372			0.356186	−	0.934415i	$-0.384077\pi$
$251$	11.2861			0.712372			0.356186	+	0.934415i	$0.384077\pi$
$252$		0			0
$253$		−	34.1003i		−	2.14387i
$254$		−	4.64476i		−	0.291438i
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	11.2550i		−	0.702066i	−0.936363	−	0.351033i	$-0.885831\pi$
$257$		−	11.2550i		−	0.702066i	0.936363	−	0.351033i	$-0.114169\pi$
$258$		0			0
$259$		0			0
$260$	2.60139	−	10.7404i	0.161332	−	0.666092i
$261$		0			0
$262$	6.16011			0.380573
$263$	−26.0470			−1.60613			−0.803064	−	0.595893i	$-0.796798\pi$
$263$	−26.0470			−1.60613			−0.803064	+	0.595893i	$0.796798\pi$
$264$		0			0
$265$	2.98608	−	12.3287i	0.183434	−	0.757345i
$266$		0			0
$267$		0			0
$268$		−	2.40838i		−	0.147115i
$269$	−8.34906			−0.509051			−0.254526	−	0.967066i	$-0.581919\pi$
$269$	−8.34906			−0.509051			−0.254526	+	0.967066i	$0.581919\pi$
$270$		0			0
$271$			20.5200i			1.24650i	0.782022	+	0.623251i	$0.214189\pi$
$271$			20.5200i			1.24650i	−0.782022	+	0.623251i	$0.785811\pi$
$272$			5.22951i			0.317086i
$273$		0			0
$274$	−11.6165			−0.701777
$275$	−10.1088	+	19.6440i	−0.609583	+	1.18458i
$276$		0			0
$277$			26.2070i			1.57463i	0.616552	+	0.787314i	$0.288529\pi$
$277$			26.2070i			1.57463i	−0.616552	+	0.787314i	$0.711471\pi$
$278$			15.1955i			0.911363i
$279$		0			0
$280$		0			0
$281$			15.9485i			0.951408i	0.879605	+	0.475704i	$0.157807\pi$
$281$			15.9485i			0.951408i	−0.879605	+	0.475704i	$0.842193\pi$
$282$		0			0
$283$	−1.03048			−0.0612558			−0.0306279	−	0.999531i	$-0.509751\pi$
$283$	−1.03048			−0.0612558			−0.0306279	+	0.999531i	$0.509751\pi$
$284$			10.4298i			0.618895i
$285$		0			0
$286$		−	21.8367i		−	1.29123i
$287$		0			0
$288$		0			0
$289$	−10.3478			−0.608694
$290$	−22.2185	−	5.38146i	−1.30472	−	0.316010i
$291$		0			0
$292$	−0.202603			−0.0118564
$293$		−	11.5472i		−	0.674593i	−0.941398	−	0.337297i	$-0.890487\pi$
$293$		−	11.5472i		−	0.674593i	0.941398	−	0.337297i	$-0.109513\pi$
$294$		0			0
$295$	6.38697	−	26.3700i	0.371864	−	1.53532i
$296$		−	9.25293i		−	0.537815i
$297$		0			0
$298$			4.18853i			0.242635i
$299$	−38.1418			−2.20580
$300$		0			0
$301$		0			0
$302$	16.3287			0.939614
$303$		0			0
$304$			4.97179i			0.285152i
$305$	14.7034	+	3.56127i	0.841917	+	0.203917i
$306$		0			0
$307$	−33.8139			−1.92986			−0.964931	−	0.262502i	$-0.915452\pi$
$307$	−33.8139			−1.92986			−0.964931	+	0.262502i	$0.915452\pi$
$308$		0			0
$309$		0			0
$310$	13.9830	+	3.38677i	0.794181	+	0.192356i
$311$	28.2780			1.60350			0.801749	−	0.597661i	$-0.203903\pi$
$311$	28.2780			1.60350			0.801749	+	0.597661i	$0.203903\pi$
$312$		0			0
$313$	−14.0532			−0.794333			−0.397166	−	0.917747i	$-0.630006\pi$
$313$	−14.0532			−0.794333			−0.397166	+	0.917747i	$0.630006\pi$
$314$	11.1873			0.631337
$315$		0			0
$316$	−0.736672			−0.0414411
$317$	−3.56582			−0.200276			−0.100138	−	0.994974i	$-0.531928\pi$
$317$	−3.56582			−0.200276			−0.100138	+	0.994974i	$0.531928\pi$
$318$		0			0
$319$	−45.1732			−2.52922
$320$	−0.526370	+	2.17323i	−0.0294250	+	0.121487i
$321$		0			0
$322$		0			0
$323$	−26.0000			−1.44668
$324$		0			0
$325$	21.9721	+	11.3069i	1.21879	+	0.627192i
$326$		−	9.35228i		−	0.517975i
$327$		0			0
$328$	−6.79316			−0.375089
$329$		0			0
$330$		0			0
$331$	−10.7995			−0.593595			−0.296798	−	0.954940i	$-0.595919\pi$
$331$	−10.7995			−0.593595			−0.296798	+	0.954940i	$0.595919\pi$
$332$			3.96971i			0.217866i
$333$		0			0
$334$		−	2.72339i		−	0.149017i
$335$	5.23397	+	1.26770i	0.285963	+	0.0692619i
$336$		0			0
$337$			3.75187i			0.204377i	0.994765	+	0.102189i	$0.0325845\pi$
$337$			3.75187i			0.204377i	−0.994765	+	0.102189i	$0.967415\pi$
$338$	−11.4247			−0.621424
$339$		0			0
$340$	−11.3649	−	2.75266i	−0.616350	−	0.149284i
$341$	28.4293			1.53954
$342$		0			0
$343$		0			0
$344$		−	2.08223i		−	0.112266i
$345$		0			0
$346$			3.90918i			0.210159i
$347$	−31.8993			−1.71245			−0.856223	−	0.516606i	$-0.827195\pi$
$347$	−31.8993			−1.71245			−0.856223	+	0.516606i	$0.827195\pi$
$348$		0			0
$349$			31.4147i			1.68159i	0.541354	+	0.840795i	$0.317912\pi$
$349$			31.4147i			1.68159i	−0.541354	+	0.840795i	$0.682088\pi$
$350$		0			0
$351$		0			0
$352$			4.41847i			0.235505i
$353$		−	32.8117i		−	1.74639i	−0.487371	−	0.873195i	$-0.662044\pi$
$353$		−	32.8117i		−	1.74639i	0.487371	−	0.873195i	$-0.337956\pi$
$354$		0			0
$355$	−22.6664	−	5.48994i	−1.20301	−	0.291376i
$356$	−6.61837			−0.350773
$357$		0			0
$358$			24.3957i			1.28935i
$359$			1.53908i			0.0812295i	0.999175	+	0.0406148i	$0.0129316\pi$
$359$			1.53908i			0.0812295i	−0.999175	+	0.0406148i	$0.987068\pi$
$360$		0			0
$361$	−5.71871			−0.300985
$362$			22.0352i			1.15814i
$363$		0			0
$364$		0			0
$365$	0.106644	−	0.440303i	0.00558201	−	0.0230465i
$366$		0			0
$367$	−18.8442			−0.983661			−0.491831	−	0.870691i	$-0.663672\pi$
$367$	−18.8442			−0.983661			−0.491831	+	0.870691i	$0.663672\pi$
$368$	7.71767			0.402311
$369$		0			0
$370$	20.1088	+	4.87046i	1.04540	+	0.253203i
$371$		0			0
$372$		0			0
$373$			6.72346i			0.348128i	0.984734	+	0.174064i	$0.0556899\pi$
$373$			6.72346i			0.348128i	−0.984734	+	0.174064i	$0.944310\pi$
$374$	−23.1065			−1.19481
$375$		0			0
$376$			7.83497i			0.404058i
$377$			50.5270i			2.60228i
$378$		0			0
$379$	−21.1124			−1.08447			−0.542235	−	0.840227i	$-0.682422\pi$
$379$	−21.1124			−1.08447			−0.542235	+	0.840227i	$0.682422\pi$
$380$	−10.8049	−	2.61700i	−0.554277	−	0.134249i
$381$		0			0
$382$			26.8532i			1.37393i
$383$		−	1.15349i		−	0.0589407i	−0.999566	−	0.0294704i	$-0.990618\pi$
$383$		−	1.15349i		−	0.0589407i	0.999566	−	0.0294704i	$-0.00938207\pi$
$384$		0			0
$385$		0			0
$386$			13.9798i			0.711554i
$387$		0			0
$388$	−14.5487			−0.738597
$389$			11.5150i			0.583833i	0.956444	+	0.291916i	$0.0942929\pi$
$389$			11.5150i			0.583833i	−0.956444	+	0.291916i	$0.905707\pi$
$390$		0			0
$391$			40.3597i			2.04108i
$392$		0			0
$393$		0			0
$394$	11.2214			0.565327
$395$	0.387762	−	1.60096i	0.0195104	−	0.0805530i
$396$		0			0
$397$	5.89542			0.295883			0.147941	−	0.988996i	$-0.452735\pi$
$397$	5.89542			0.295883			0.147941	+	0.988996i	$0.452735\pi$
$398$		−	1.80074i		−	0.0902628i
$399$		0			0
$400$	−4.44587	−	2.28785i	−0.222293	−	0.114392i
$401$		−	19.1472i		−	0.956166i	−0.878315	−	0.478083i	$-0.841332\pi$
$401$		−	19.1472i		−	0.956166i	0.878315	−	0.478083i	$-0.158668\pi$
$402$		0			0
$403$		−	31.7987i		−	1.58401i
$404$	−5.16164			−0.256801
$405$		0			0
$406$		0			0
$407$	40.8838			2.02654
$408$		0			0
$409$			19.0702i			0.942962i	0.881876	+	0.471481i	$0.156280\pi$
$409$			19.0702i			0.942962i	−0.881876	+	0.471481i	$0.843720\pi$
$410$	3.57572	−	14.7631i	0.176592	−	0.729098i
$411$		0			0
$412$	4.62070			0.227646
$413$		0			0
$414$		0			0
$415$	−8.62709	−	2.08954i	−0.423487	−	0.102571i
$416$	4.94214			0.242308
$417$		0			0
$418$	−21.9677			−1.07448
$419$	−2.48991			−0.121640			−0.0608199	−	0.998149i	$-0.519372\pi$
$419$	−2.48991			−0.121640			−0.0608199	+	0.998149i	$0.519372\pi$
$420$		0			0
$421$	7.61083			0.370929			0.185465	−	0.982651i	$-0.440621\pi$
$421$	7.61083			0.370929			0.185465	+	0.982651i	$0.440621\pi$
$422$	−1.70144			−0.0828248
$423$		0			0
$424$	5.67297			0.275504
$425$	11.9643	−	23.2497i	0.580355	−	1.12778i
$426$		0			0
$427$		0			0
$428$	−10.1677			−0.491472
$429$		0			0
$430$	4.52517	+	1.09603i	0.218223	+	0.0528550i
$431$		−	9.28436i		−	0.447212i	−0.974680	−	0.223606i	$-0.928217\pi$
$431$		−	9.28436i		−	0.447212i	0.974680	−	0.223606i	$-0.0717828\pi$
$432$		0			0
$433$	−25.5306			−1.22692			−0.613460	−	0.789726i	$-0.710223\pi$
$433$	−25.5306			−1.22692			−0.613460	+	0.789726i	$0.710223\pi$
$434$		0			0
$435$		0			0
$436$	−9.18677			−0.439966
$437$			38.3706i			1.83552i
$438$		0			0
$439$			9.10460i			0.434539i	0.976112	+	0.217269i	$0.0697150\pi$
$439$			9.10460i			0.434539i	−0.976112	+	0.217269i	$0.930285\pi$
$440$	−9.60237	−	2.32575i	−0.457775	−	0.110876i
$441$		0			0
$442$			25.8450i			1.22932i
$443$	21.6070			1.02658			0.513289	−	0.858216i	$-0.328427\pi$
$443$	21.6070			1.02658			0.513289	+	0.858216i	$0.328427\pi$
$444$		0			0
$445$	3.48371	−	14.3833i	0.165144	−	0.681832i
$446$	−3.22392			−0.152657
$447$		0			0
$448$		0			0
$449$			13.3403i			0.629566i	0.949164	+	0.314783i	$0.101932\pi$
$449$			13.3403i			0.629566i	−0.949164	+	0.314783i	$0.898068\pi$
$450$		0			0
$451$		−	30.0154i		−	1.41337i
$452$	7.19926			0.338625
$453$		0			0
$454$			2.34174i			0.109904i
$455$		0			0
$456$		0			0
$457$		−	31.7813i		−	1.48667i	−0.668921	−	0.743334i	$-0.733243\pi$
$457$		−	31.7813i		−	1.48667i	0.668921	−	0.743334i	$-0.266757\pi$
$458$			11.6386i			0.543837i
$459$		0			0
$460$	−4.06235	+	16.7723i	−0.189408	+	0.782012i
$461$	4.61323			0.214860			0.107430	−	0.994213i	$-0.465738\pi$
$461$	4.61323			0.214860			0.107430	+	0.994213i	$0.465738\pi$
$462$		0			0
$463$		−	14.0850i		−	0.654587i	−0.944923	−	0.327293i	$-0.893863\pi$
$463$		−	14.0850i		−	0.654587i	0.944923	−	0.327293i	$-0.106137\pi$
$464$		−	10.2237i		−	0.474624i
$465$		0			0
$466$	0.0869816			0.00402935
$467$			39.9154i			1.84707i	0.383520	+	0.923533i	$0.374712\pi$
$467$			39.9154i			1.84707i	−0.383520	+	0.923533i	$0.625288\pi$
$468$		0			0
$469$		0			0
$470$	−17.0272	−	4.12409i	−0.785406	−	0.190230i
$471$		0			0
$472$	12.1340			0.558512
$473$	9.20029			0.423030
$474$		0			0
$475$	11.3747	−	22.1039i	0.521907	−	1.01420i
$476$		0			0
$477$		0			0
$478$		−	13.6099i		−	0.622502i
$479$	−22.3228			−1.01995			−0.509977	−	0.860188i	$-0.670346\pi$
$479$	−22.3228			−1.01995			−0.509977	+	0.860188i	$0.670346\pi$
$480$		0			0
$481$		−	45.7293i		−	2.08507i
$482$			5.13807i			0.234033i
$483$		0			0
$484$	−8.52291			−0.387405
$485$	7.65798	−	31.6176i	0.347731	−	1.43568i
$486$		0			0
$487$		−	41.2767i		−	1.87043i	−0.354085	−	0.935213i	$-0.615208\pi$
$487$		−	41.2767i		−	1.87043i	0.354085	−	0.935213i	$-0.384792\pi$
$488$			6.76571i			0.306269i
$489$		0			0
$490$		0			0
$491$		−	14.6041i		−	0.659075i	−0.944143	−	0.329537i	$-0.893107\pi$
$491$		−	14.6041i		−	0.659075i	0.944143	−	0.329537i	$-0.106893\pi$
$492$		0			0
$493$	53.4651			2.40795
$494$			24.5713i			1.10551i
$495$		0			0
$496$			6.43420i			0.288904i
$497$		0			0
$498$		0			0
$499$	13.6787			0.612342			0.306171	−	0.951976i	$-0.400952\pi$
$499$	13.6787			0.612342			0.306171	+	0.951976i	$0.400952\pi$
$500$	7.31220	−	8.45765i	0.327011	−	0.378237i
$501$		0			0
$502$	−11.2861			−0.503723
$503$		−	7.92976i		−	0.353570i	−0.984249	−	0.176785i	$-0.943430\pi$
$503$		−	7.92976i		−	0.353570i	0.984249	−	0.176785i	$-0.0565698\pi$
$504$		0			0
$505$	2.71693	−	11.2174i	0.120902	−	0.499169i
$506$			34.1003i			1.51594i
$507$		0			0
$508$			4.64476i			0.206078i
$509$	3.47370			0.153969			0.0769845	−	0.997032i	$-0.475471\pi$
$509$	3.47370			0.153969			0.0769845	+	0.997032i	$0.475471\pi$
$510$		0			0
$511$		0			0
$512$	−1.00000			−0.0441942
$513$		0			0
$514$			11.2550i			0.496435i
$515$	−2.43220	+	10.0419i	−0.107176	+	0.442497i
$516$		0			0
$517$	−34.6186			−1.52252
$518$		0			0
$519$		0			0
$520$	−2.60139	+	10.7404i	−0.114079	+	0.470998i
$521$	18.7795			0.822746			0.411373	−	0.911467i	$-0.365049\pi$
$521$	18.7795			0.822746			0.411373	+	0.911467i	$0.365049\pi$
$522$		0			0
$523$	−12.0104			−0.525176			−0.262588	−	0.964908i	$-0.584576\pi$
$523$	−12.0104			−0.525176			−0.262588	+	0.964908i	$0.584576\pi$
$524$	−6.16011			−0.269106
$525$		0			0
$526$	26.0470			1.13570
$527$	−33.6477			−1.46572
$528$		0			0
$529$	36.5624			1.58967
$530$	−2.98608	+	12.3287i	−0.129707	+	0.535523i
$531$		0			0
$532$		0			0
$533$	−33.5727			−1.45420
$534$		0			0
$535$	5.35195	−	22.0967i	0.231385	−	0.955322i
$536$			2.40838i			0.104026i
$537$		0			0
$538$	8.34906			0.359954
$539$		0			0
$540$		0			0
$541$	−4.18448			−0.179905			−0.0899523	−	0.995946i	$-0.528671\pi$
$541$	−4.18448			−0.179905			−0.0899523	+	0.995946i	$0.528671\pi$
$542$		−	20.5200i		−	0.881410i
$543$		0			0
$544$		−	5.22951i		−	0.224214i
$545$	4.83564	−	19.9650i	0.207136	−	0.855206i
$546$		0			0
$547$		−	21.8675i		−	0.934986i	−0.883997	−	0.467493i	$-0.845157\pi$
$547$		−	21.8675i		−	0.934986i	0.883997	−	0.467493i	$-0.154843\pi$
$548$	11.6165			0.496231
$549$		0			0
$550$	10.1088	−	19.6440i	0.431041	−	0.837621i
$551$	50.8302			2.16544
$552$		0			0
$553$		0			0
$554$		−	26.2070i		−	1.11343i
$555$		0			0
$556$		−	15.1955i		−	0.644431i
$557$	−21.3530			−0.904755			−0.452377	−	0.891827i	$-0.649424\pi$
$557$	−21.3530			−0.904755			−0.452377	+	0.891827i	$0.649424\pi$
$558$		0			0
$559$		−	10.2907i		−	0.435250i
$560$		0			0
$561$		0			0
$562$		−	15.9485i		−	0.672747i
$563$		−	15.5975i		−	0.657356i	−0.944442	−	0.328678i	$-0.893397\pi$
$563$		−	15.5975i		−	0.657356i	0.944442	−	0.328678i	$-0.106603\pi$
$564$		0			0
$565$	−3.78947	+	15.6457i	−0.159424	+	0.658218i
$566$	1.03048			0.0433144
$567$		0			0
$568$		−	10.4298i		−	0.437625i
$569$			19.9293i			0.835481i	0.908566	+	0.417741i	$0.137178\pi$
$569$			19.9293i			0.835481i	−0.908566	+	0.417741i	$0.862822\pi$
$570$		0			0
$571$	−36.2857			−1.51851			−0.759255	−	0.650793i	$-0.774436\pi$
$571$	−36.2857			−1.51851			−0.759255	+	0.650793i	$0.774436\pi$
$572$			21.8367i			0.913039i
$573$		0			0
$574$		0			0
$575$	−34.3117	−	17.6569i	−1.43090	−	0.736342i
$576$		0			0
$577$	−27.4529			−1.14288			−0.571439	−	0.820645i	$-0.693615\pi$
$577$	−27.4529			−1.14288			−0.571439	+	0.820645i	$0.693615\pi$
$578$	10.3478			0.430412
$579$		0			0
$580$	22.2185	+	5.38146i	0.922573	+	0.223453i
$581$		0			0
$582$		0			0
$583$			25.0659i			1.03812i
$584$	0.202603			0.00838377
$585$		0			0
$586$			11.5472i			0.477010i
$587$			21.0716i			0.869719i	0.900498	+	0.434859i	$0.143202\pi$
$587$			21.0716i			0.869719i	−0.900498	+	0.434859i	$0.856798\pi$
$588$		0			0
$589$	−31.9895			−1.31810
$590$	−6.38697	+	26.3700i	−0.262947	+	1.08563i
$591$		0			0
$592$			9.25293i			0.380293i
$593$			5.31080i			0.218088i	0.994037	+	0.109044i	$0.0347790\pi$
$593$			5.31080i			0.218088i	−0.994037	+	0.109044i	$0.965221\pi$
$594$		0			0
$595$		0			0
$596$		−	4.18853i		−	0.171569i
$597$		0			0
$598$	38.1418			1.55973
$599$			6.56822i			0.268370i	0.990956	+	0.134185i	$0.0428417\pi$
$599$			6.56822i			0.268370i	−0.990956	+	0.134185i	$0.957158\pi$
$600$		0			0
$601$		−	22.0755i		−	0.900479i	−0.892908	−	0.450239i	$-0.851339\pi$
$601$		−	22.0755i		−	0.900479i	0.892908	−	0.450239i	$-0.148661\pi$
$602$		0			0
$603$		0			0
$604$	−16.3287			−0.664407
$605$	4.48621	−	18.5223i	0.182390	−	0.753037i
$606$		0			0
$607$	−0.461776			−0.0187429			−0.00937145	−	0.999956i	$-0.502983\pi$
$607$	−0.461776			−0.0187429			−0.00937145	+	0.999956i	$0.502983\pi$
$608$		−	4.97179i		−	0.201633i
$609$		0			0
$610$	−14.7034	−	3.56127i	−0.595325	−	0.144191i
$611$			38.7215i			1.56650i
$612$		0			0
$613$			34.1411i			1.37895i	0.724312	+	0.689473i	$0.242158\pi$
$613$			34.1411i			1.37895i	−0.724312	+	0.689473i	$0.757842\pi$
$614$	33.8139			1.36462
$615$		0			0
$616$		0			0
$617$	−27.1936			−1.09477			−0.547387	−	0.836880i	$-0.684377\pi$
$617$	−27.1936			−1.09477			−0.547387	+	0.836880i	$0.684377\pi$
$618$		0			0
$619$			1.75549i			0.0705591i	0.999377	+	0.0352795i	$0.0112322\pi$
$619$			1.75549i			0.0705591i	−0.999377	+	0.0352795i	$0.988768\pi$
$620$	−13.9830	−	3.38677i	−0.561571	−	0.136016i
$621$		0			0
$622$	−28.2780			−1.13384
$623$		0			0
$624$		0			0
$625$	14.5315	+	20.3429i	0.581260	+	0.813718i
$626$	14.0532			0.561678
$627$		0			0
$628$	−11.1873			−0.446423
$629$	−48.3883			−1.92937
$630$		0			0
$631$	30.6710			1.22099			0.610496	−	0.792019i	$-0.290970\pi$
$631$	30.6710			1.22099			0.610496	+	0.792019i	$0.290970\pi$
$632$	0.736672			0.0293033
$633$		0			0
$634$	3.56582			0.141617
$635$	−10.0941	−	2.44486i	−0.400573	−	0.0970213i
$636$		0			0
$637$		0			0
$638$	45.1732			1.78843
$639$		0			0
$640$	0.526370	−	2.17323i	0.0208066	−	0.0859045i
$641$		−	14.2174i		−	0.561553i	−0.959773	−	0.280777i	$-0.909408\pi$
$641$		−	14.2174i		−	0.561553i	0.959773	−	0.280777i	$-0.0905920\pi$
$642$		0			0
$643$	40.4526			1.59529			0.797647	−	0.603124i	$-0.206078\pi$
$643$	40.4526			1.59529			0.797647	+	0.603124i	$0.206078\pi$
$644$		0			0
$645$		0			0
$646$	26.0000			1.02296
$647$			12.9418i			0.508793i	0.967100	+	0.254397i	$0.0818768\pi$
$647$			12.9418i			0.508793i	−0.967100	+	0.254397i	$0.918123\pi$
$648$		0			0
$649$			53.6137i			2.10452i
$650$	−21.9721	−	11.3069i	−0.861817	−	0.443492i
$651$		0			0
$652$			9.35228i			0.366263i
$653$	−20.3866			−0.797791			−0.398895	−	0.916996i	$-0.630606\pi$
$653$	−20.3866			−0.797791			−0.398895	+	0.916996i	$0.630606\pi$
$654$		0			0
$655$	3.24250	−	13.3873i	0.126695	−	0.523087i
$656$	6.79316			0.265228
$657$		0			0
$658$		0			0
$659$		−	10.4950i		−	0.408829i	−0.978884	−	0.204414i	$-0.934471\pi$
$659$		−	10.4950i		−	0.408829i	0.978884	−	0.204414i	$-0.0655290\pi$
$660$		0			0
$661$		−	3.03266i		−	0.117957i	−0.998259	−	0.0589783i	$-0.981216\pi$
$661$		−	3.03266i		−	0.117957i	0.998259	−	0.0589783i	$-0.0187843\pi$
$662$	10.7995			0.419735
$663$		0			0
$664$		−	3.96971i		−	0.154055i
$665$		0			0
$666$		0			0
$667$		−	78.9033i		−	3.05515i
$668$			2.72339i			0.105371i
$669$		0			0
$670$	−5.23397	−	1.26770i	−0.202206	−	0.0489756i
$671$	−29.8941			−1.15405
$672$		0			0
$673$		−	11.2777i		−	0.434724i	−0.976091	−	0.217362i	$-0.930255\pi$
$673$		−	11.2777i		−	0.434724i	0.976091	−	0.217362i	$-0.0697452\pi$
$674$		−	3.75187i		−	0.144517i
$675$		0			0
$676$	11.4247			0.439413
$677$		−	39.9757i		−	1.53639i	−0.640216	−	0.768195i	$-0.721155\pi$
$677$		−	39.9757i		−	1.53639i	0.640216	−	0.768195i	$-0.278845\pi$
$678$		0			0
$679$		0			0
$680$	11.3649	+	2.75266i	0.435826	+	0.105560i
$681$		0			0
$682$	−28.4293			−1.08862
$683$	6.07806			0.232571			0.116285	−	0.993216i	$-0.462901\pi$
$683$	6.07806			0.232571			0.116285	+	0.993216i	$0.462901\pi$
$684$		0			0
$685$	−6.11456	+	25.2453i	−0.233626	+	0.964573i
$686$		0			0
$687$		0			0
$688$			2.08223i			0.0793844i
$689$	28.0366			1.06811
$690$		0			0
$691$		−	28.5091i		−	1.08454i	−0.840205	−	0.542269i	$-0.817565\pi$
$691$		−	28.5091i		−	1.08454i	0.840205	−	0.542269i	$-0.182435\pi$
$692$		−	3.90918i		−	0.148605i
$693$		0			0
$694$	31.8993			1.21088
$695$	33.0232	+	7.99843i	1.25264	+	0.303398i
$696$		0			0
$697$			35.5249i			1.34560i
$698$		−	31.4147i		−	1.18906i
$699$		0			0
$700$		0			0
$701$			17.1007i			0.645885i	0.946419	+	0.322943i	$0.104672\pi$
$701$			17.1007i			0.645885i	−0.946419	+	0.322943i	$0.895328\pi$
$702$		0			0
$703$	−46.0036			−1.73506
$704$		−	4.41847i		−	0.166528i
$705$		0			0
$706$			32.8117i			1.23488i
$707$		0			0
$708$		0			0
$709$	29.8197			1.11990			0.559950	−	0.828526i	$-0.310820\pi$
$709$	29.8197			1.11990			0.559950	+	0.828526i	$0.310820\pi$
$710$	22.6664	+	5.48994i	0.850654	+	0.206034i
$711$		0			0
$712$	6.61837			0.248034
$713$			49.6570i			1.85967i
$714$		0			0
$715$	−47.4562	−	11.4942i	−1.77476	−	0.429858i
$716$		−	24.3957i		−	0.911711i
$717$		0			0
$718$		−	1.53908i		−	0.0574379i
$719$	−13.6376			−0.508597			−0.254298	−	0.967126i	$-0.581845\pi$
$719$	−13.6376			−0.508597			−0.254298	+	0.967126i	$0.581845\pi$
$720$		0			0
$721$		0			0
$722$	5.71871			0.212828
$723$		0			0
$724$		−	22.0352i		−	0.818930i
$725$	−23.3903	+	45.4533i	−0.868694	+	1.68809i
$726$		0			0
$727$	−18.3907			−0.682075			−0.341037	−	0.940050i	$-0.610778\pi$
$727$	−18.3907			−0.682075			−0.341037	+	0.940050i	$0.610778\pi$
$728$		0			0
$729$		0			0
$730$	−0.106644	+	0.440303i	−0.00394708	+	0.0162964i
$731$	−10.8891			−0.402747
$732$		0			0
$733$	43.2037			1.59577			0.797883	−	0.602813i	$-0.205953\pi$
$733$	43.2037			1.59577			0.797883	+	0.602813i	$0.205953\pi$
$734$	18.8442			0.695553
$735$		0			0
$736$	−7.71767			−0.284477
$737$	−10.6414			−0.391980
$738$		0			0
$739$	37.1476			1.36650			0.683249	−	0.730186i	$-0.260566\pi$
$739$	37.1476			1.36650			0.683249	+	0.730186i	$0.260566\pi$
$740$	−20.1088	−	4.87046i	−0.739212	−	0.179042i
$741$		0			0
$742$		0			0
$743$	9.11107			0.334253			0.167126	−	0.985935i	$-0.446551\pi$
$743$	9.11107			0.334253			0.167126	+	0.985935i	$0.446551\pi$
$744$		0			0
$745$	9.10265	+	2.20472i	0.333495	+	0.0807747i
$746$		−	6.72346i		−	0.246163i
$747$		0			0
$748$	23.1065			0.844856
$749$		0			0
$750$		0			0
$751$	24.1898			0.882698			0.441349	−	0.897336i	$-0.354500\pi$
$751$	24.1898			0.882698			0.441349	+	0.897336i	$0.354500\pi$
$752$		−	7.83497i		−	0.285712i
$753$		0			0
$754$		−	50.5270i		−	1.84009i
$755$	8.59496	−	35.4861i	0.312803	−	1.29147i
$756$		0			0
$757$			46.1258i			1.67647i	0.545310	+	0.838235i	$0.316412\pi$
$757$			46.1258i			1.67647i	−0.545310	+	0.838235i	$0.683588\pi$
$758$	21.1124			0.766836
$759$		0			0
$760$	10.8049	+	2.61700i	0.391933	+	0.0949287i
$761$	−32.8441			−1.19060			−0.595299	−	0.803504i	$-0.702966\pi$
$761$	−32.8441			−1.19060			−0.595299	+	0.803504i	$0.702966\pi$
$762$		0			0
$763$		0			0
$764$		−	26.8532i		−	0.971514i
$765$		0			0
$766$			1.15349i			0.0416774i
$767$	59.9678			2.16531
$768$		0			0
$769$			41.7778i			1.50655i	0.657708	+	0.753273i	$0.271526\pi$
$769$			41.7778i			1.50655i	−0.657708	+	0.753273i	$0.728474\pi$
$770$		0			0
$771$		0			0
$772$		−	13.9798i		−	0.503145i
$773$		−	19.8712i		−	0.714717i	−0.933967	−	0.357359i	$-0.883677\pi$
$773$		−	19.8712i		−	0.714717i	0.933967	−	0.357359i	$-0.116323\pi$
$774$		0			0
$775$	14.7205	−	28.6056i	0.528775	−	1.02754i
$776$	14.5487			0.522267
$777$		0			0
$778$		−	11.5150i		−	0.412832i
$779$			33.7742i			1.21009i
$780$		0			0
$781$	46.0838			1.64901
$782$		−	40.3597i		−	1.44326i
$783$		0			0
$784$		0			0
$785$	5.88867	−	24.3126i	0.210176	−	0.867755i
$786$		0			0
$787$	−39.5693			−1.41049			−0.705246	−	0.708963i	$-0.749163\pi$
$787$	−39.5693			−1.41049			−0.705246	+	0.708963i	$0.749163\pi$
$788$	−11.2214			−0.399746
$789$		0			0
$790$	−0.387762	+	1.60096i	−0.0137960	+	0.0569596i
$791$		0			0
$792$		0			0
$793$			33.4371i			1.18739i
$794$	−5.89542			−0.209221
$795$		0			0
$796$			1.80074i			0.0638254i
$797$			37.8029i			1.33905i	0.742790	+	0.669524i	$0.233502\pi$
$797$			37.8029i			1.33905i	−0.742790	+	0.669524i	$0.766498\pi$
$798$		0			0
$799$	40.9731			1.44952
$800$	4.44587	+	2.28785i	0.157185	+	0.0808876i
$801$		0			0
$802$			19.1472i			0.676112i
$803$			0.895196i			0.0315908i
$804$		0			0
$805$		0			0
$806$			31.7987i			1.12006i
$807$		0			0
$808$	5.16164			0.181586
$809$			12.9081i			0.453826i	0.973915	+	0.226913i	$0.0728633\pi$
$809$			12.9081i			0.453826i	−0.973915	+	0.226913i	$0.927137\pi$
$810$		0			0
$811$		−	3.57478i		−	0.125527i	−0.998028	−	0.0627637i	$-0.980009\pi$
$811$		−	3.57478i		−	0.125527i	0.998028	−	0.0627637i	$-0.0199914\pi$
$812$		0			0
$813$		0			0
$814$	−40.8838			−1.43298
$815$	−20.3247	−	4.92276i	−0.711942	−	0.172437i
$816$		0			0
$817$	−10.3524			−0.362186
$818$		−	19.0702i		−	0.666775i
$819$		0			0
$820$	−3.57572	+	14.7631i	−0.124869	+	0.515550i
$821$			41.2899i			1.44103i	0.693440	+	0.720514i	$0.256094\pi$
$821$			41.2899i			1.44103i	−0.693440	+	0.720514i	$0.743906\pi$
$822$		0			0
$823$		−	12.5211i		−	0.436459i	−0.975897	−	0.218229i	$-0.929972\pi$
$823$		−	12.5211i		−	0.436459i	0.975897	−	0.218229i	$-0.0700281\pi$
$824$	−4.62070			−0.160970
$825$		0			0
$826$		0			0
$827$	28.4665			0.989877			0.494938	−	0.868928i	$-0.335191\pi$
$827$	28.4665			0.989877			0.494938	+	0.868928i	$0.335191\pi$
$828$		0			0
$829$		−	42.3188i		−	1.46979i	−0.678179	−	0.734897i	$-0.737231\pi$
$829$		−	42.3188i		−	1.46979i	0.678179	−	0.734897i	$-0.262769\pi$
$830$	8.62709	+	2.08954i	0.299451	+	0.0725288i
$831$		0			0
$832$	−4.94214			−0.171338
$833$		0			0
$834$		0			0
$835$	−5.91855	−	1.43351i	−0.204820	−	0.0496086i
$836$	21.9677			0.759770
$837$		0			0
$838$	2.48991			0.0860124
$839$	−48.5307			−1.67547			−0.837733	−	0.546080i	$-0.816120\pi$
$839$	−48.5307			−1.67547			−0.837733	+	0.546080i	$0.816120\pi$
$840$		0			0
$841$	−75.5244			−2.60429
$842$	−7.61083			−0.262287
$843$		0			0
$844$	1.70144			0.0585659
$845$	−6.01364	+	24.8286i	−0.206876	+	0.854130i
$846$		0			0
$847$		0			0
$848$	−5.67297			−0.194811
$849$		0			0
$850$	−11.9643	+	23.2497i	−0.410373	+	0.797459i
$851$			71.4110i			2.44794i
$852$		0			0
$853$	7.69534			0.263483			0.131742	−	0.991284i	$-0.457943\pi$
$853$	7.69534			0.263483			0.131742	+	0.991284i	$0.457943\pi$
$854$		0			0
$855$		0			0
$856$	10.1677			0.347523
$857$		−	6.59617i		−	0.225321i	−0.993634	−	0.112660i	$-0.964063\pi$
$857$		−	6.59617i		−	0.225321i	0.993634	−	0.112660i	$-0.0359372\pi$
$858$		0			0
$859$		−	5.46161i		−	0.186348i	−0.995650	−	0.0931738i	$-0.970299\pi$
$859$		−	5.46161i		−	0.186348i	0.995650	−	0.0931738i	$-0.0297012\pi$
$860$	−4.52517	−	1.09603i	−0.154307	−	0.0373741i
$861$		0			0
$862$			9.28436i			0.316226i
$863$	−46.8283			−1.59405			−0.797027	−	0.603943i	$-0.793595\pi$
$863$	−46.8283			−1.59405			−0.797027	+	0.603943i	$0.793595\pi$
$864$		0			0
$865$	8.49555	+	2.05768i	0.288858	+	0.0699631i
$866$	25.5306			0.867564
$867$		0			0
$868$		0			0
$869$			3.25497i			0.110417i
$870$		0			0
$871$			11.9026i			0.403303i
$872$	9.18677			0.311103
$873$		0			0
$874$		−	38.3706i		−	1.29791i
$875$		0			0
$876$		0			0
$877$			18.9375i			0.639474i	0.947506	+	0.319737i	$0.103595\pi$
$877$			18.9375i			0.639474i	−0.947506	+	0.319737i	$0.896405\pi$
$878$		−	9.10460i		−	0.307265i
$879$		0			0
$880$	9.60237	+	2.32575i	0.323696	+	0.0784011i
$881$	2.20766			0.0743778			0.0371889	−	0.999308i	$-0.488160\pi$
$881$	2.20766			0.0743778			0.0371889	+	0.999308i	$0.488160\pi$
$882$		0			0
$883$		−	10.4582i		−	0.351948i	−0.984395	−	0.175974i	$-0.943693\pi$
$883$		−	10.4582i		−	0.351948i	0.984395	−	0.175974i	$-0.0563074\pi$
$884$		−	25.8450i		−	0.869261i
$885$		0			0
$886$	−21.6070			−0.725900
$887$		−	12.6894i		−	0.426068i	−0.977045	−	0.213034i	$-0.931666\pi$
$887$		−	12.6894i		−	0.426068i	0.977045	−	0.213034i	$-0.0683345\pi$
$888$		0			0
$889$		0			0
$890$	−3.48371	+	14.3833i	−0.116774	+	0.482128i
$891$		0			0
$892$	3.22392			0.107945
$893$	38.9538			1.30354
$894$		0			0
$895$	53.0176	+	12.8412i	1.77218	+	0.429233i
$896$		0			0
$897$		0			0
$898$		−	13.3403i		−	0.445170i
$899$	65.7814			2.19393
$900$		0			0
$901$		−	29.6669i		−	0.988347i
$902$			30.0154i			0.999403i
$903$		0			0
$904$	−7.19926			−0.239444
$905$	47.8875	+	11.5986i	1.59183	+	0.385552i
$906$		0			0
$907$		−	29.8458i		−	0.991013i	−0.868604	−	0.495506i	$-0.834982\pi$
$907$		−	29.8458i		−	0.991013i	0.868604	−	0.495506i	$-0.165018\pi$
$908$		−	2.34174i		−	0.0777135i
$909$		0			0
$910$		0			0
$911$			43.2244i			1.43209i	0.698055	+	0.716044i	$0.254049\pi$
$911$			43.2244i			1.43209i	−0.698055	+	0.716044i	$0.745951\pi$
$912$		0			0
$913$	17.5401			0.580491
$914$			31.7813i			1.05123i
$915$		0			0
$916$		−	11.6386i		−	0.384551i
$917$		0			0
$918$		0			0
$919$	8.99346			0.296667			0.148333	−	0.988937i	$-0.452609\pi$
$919$	8.99346			0.296667			0.148333	+	0.988937i	$0.452609\pi$
$920$	4.06235	−	16.7723i	0.133932	−	0.552966i
$921$		0			0
$922$	−4.61323			−0.151929
$923$		−	51.5455i		−	1.69664i
$924$		0			0
$925$	21.1693	−	41.1373i	0.696042	−	1.35259i
$926$			14.0850i			0.462863i
$927$		0			0
$928$			10.2237i			0.335610i
$929$	−1.64006			−0.0538088			−0.0269044	−	0.999638i	$-0.508565\pi$
$929$	−1.64006			−0.0538088			−0.0269044	+	0.999638i	$0.508565\pi$
$930$		0			0
$931$		0			0
$932$	−0.0869816			−0.00284918
$933$		0			0
$934$		−	39.9154i		−	1.30607i
$935$	−12.1626	+	50.2157i	−0.397758	+	1.64223i
$936$		0			0
$937$	−28.6237			−0.935095			−0.467547	−	0.883968i	$-0.654862\pi$
$937$	−28.6237			−0.935095			−0.467547	+	0.883968i	$0.654862\pi$
$938$		0			0
$939$		0			0
$940$	17.0272	+	4.12409i	0.555366	+	0.134513i
$941$	47.0222			1.53288			0.766440	−	0.642316i	$-0.222026\pi$
$941$	47.0222			1.53288			0.766440	+	0.642316i	$0.222026\pi$
$942$		0			0
$943$	52.4274			1.70727
$944$	−12.1340			−0.394928
$945$		0			0
$946$	−9.20029			−0.299127
$947$	−12.0543			−0.391714			−0.195857	−	0.980632i	$-0.562749\pi$
$947$	−12.0543			−0.391714			−0.195857	+	0.980632i	$0.562749\pi$
$948$		0			0
$949$	1.00129			0.0325033
$950$	−11.3747	+	22.1039i	−0.369044	+	0.717146i
$951$		0			0
$952$		0			0
$953$	−33.3751			−1.08113			−0.540563	−	0.841304i	$-0.681789\pi$
$953$	−33.3751			−1.08113			−0.540563	+	0.841304i	$0.681789\pi$
$954$		0			0
$955$	58.3582	+	14.1347i	1.88843	+	0.457388i
$956$			13.6099i			0.440175i
$957$		0			0
$958$	22.3228			0.721217
$959$		0			0
$960$		0			0
$961$	−10.3989			−0.335449
$962$			45.7293i			1.47437i
$963$		0			0
$964$		−	5.13807i		−	0.165486i
$965$	30.3814	+	7.35856i	0.978011	+	0.236880i
$966$		0			0
$967$			60.1284i			1.93360i	0.255535	+	0.966800i	$0.417749\pi$
$967$			60.1284i			1.93360i	−0.255535	+	0.966800i	$0.582251\pi$
$968$	8.52291			0.273937
$969$		0			0
$970$	−7.65798	+	31.6176i	−0.245883	+	1.01518i
$971$	18.8075			0.603562			0.301781	−	0.953377i	$-0.402419\pi$
$971$	18.8075			0.603562			0.301781	+	0.953377i	$0.402419\pi$
$972$		0			0
$973$		0			0
$974$			41.2767i			1.32259i
$975$		0			0
$976$		−	6.76571i		−	0.216565i
$977$	44.7033			1.43019			0.715093	−	0.699029i	$-0.246384\pi$
$977$	44.7033			1.43019			0.715093	+	0.699029i	$0.246384\pi$
$978$		0			0
$979$			29.2431i			0.934614i
$980$		0			0
$981$		0			0
$982$			14.6041i			0.466036i
$983$			28.9964i			0.924842i	0.886660	+	0.462421i	$0.153019\pi$
$983$			28.9964i			0.924842i	−0.886660	+	0.462421i	$0.846981\pi$
$984$		0			0
$985$	5.90662	−	24.3867i	0.188201	−	0.777026i
$986$	−53.4651			−1.70267
$987$		0			0
$988$		−	24.5713i		−	0.781717i
$989$			16.0700i			0.510996i
$990$		0			0
$991$	58.2764			1.85121			0.925605	−	0.378491i	$-0.123557\pi$
$991$	58.2764			1.85121			0.925605	+	0.378491i	$0.123557\pi$
$992$		−	6.43420i		−	0.204286i
$993$		0			0
$994$		0			0
$995$	−3.91342	−	0.947854i	−0.124064	−	0.0300490i
$996$		0			0
$997$	2.40483			0.0761618			0.0380809	−	0.999275i	$-0.487876\pi$
$997$	2.40483			0.0761618			0.0380809	+	0.999275i	$0.487876\pi$
$998$	−13.6787			−0.432992
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.d.c.4409.12	yes	24
3.2	odd	2		4410.2.d.d.4409.13	yes	24
5.4	even	2		4410.2.d.d.4409.11	yes	24
7.6	odd	2	inner	4410.2.d.c.4409.13	yes	24
15.14	odd	2	inner	4410.2.d.c.4409.14	yes	24
21.20	even	2		4410.2.d.d.4409.12	yes	24
35.34	odd	2		4410.2.d.d.4409.14	yes	24
105.104	even	2	inner	4410.2.d.c.4409.11	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4410.2.d.c.4409.11	✓	24	105.104	even	2	inner
4410.2.d.c.4409.12	yes	24	1.1	even	1	trivial
4410.2.d.c.4409.13	yes	24	7.6	odd	2	inner
4410.2.d.c.4409.14	yes	24	15.14	odd	2	inner
4410.2.d.d.4409.11	yes	24	5.4	even	2
4410.2.d.d.4409.12	yes	24	21.20	even	2
4410.2.d.d.4409.13	yes	24	3.2	odd	2
4410.2.d.d.4409.14	yes	24	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-0.526370 + 2.17323i) q^{5} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +(-0.526370 + 2.17323i) q^{5} -1.00000 q^{8} +(0.526370 - 2.17323i) q^{10} -4.41847i q^{11} -4.94214 q^{13} +1.00000 q^{16} +5.22951i q^{17} +4.97179i q^{19} +(-0.526370 + 2.17323i) q^{20} +4.41847i q^{22} +7.71767 q^{23} +(-4.44587 - 2.28785i) q^{25} +4.94214 q^{26} -10.2237i q^{29} +6.43420i q^{31} -1.00000 q^{32} -5.22951i q^{34} +9.25293i q^{37} -4.97179i q^{38} +(0.526370 - 2.17323i) q^{40} +6.79316 q^{41} +2.08223i q^{43} -4.41847i q^{44} -7.71767 q^{46} -7.83497i q^{47} +(4.44587 + 2.28785i) q^{50} -4.94214 q^{52} -5.67297 q^{53} +(9.60237 + 2.32575i) q^{55} +10.2237i q^{58} -12.1340 q^{59} -6.76571i q^{61} -6.43420i q^{62} +1.00000 q^{64} +(2.60139 - 10.7404i) q^{65} -2.40838i q^{67} +5.22951i q^{68} +10.4298i q^{71} -0.202603 q^{73} -9.25293i q^{74} +4.97179i q^{76} -0.736672 q^{79} +(-0.526370 + 2.17323i) q^{80} -6.79316 q^{82} +3.96971i q^{83} +(-11.3649 - 2.75266i) q^{85} -2.08223i q^{86} +4.41847i q^{88} -6.61837 q^{89} +7.71767 q^{92} +7.83497i q^{94} +(-10.8049 - 2.61700i) q^{95} -14.5487 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8}+O(q^{10}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8	\( 24 q - 24 q^{2} + 24 q^{4} - 24 q^{8} + 24 q^{16} + 32 q^{23} - 16 q^{25} - 24 q^{32} - 32 q^{46} + 16 q^{50} - 32 q^{53} + 24 q^{64} - 32 q^{85} + 32 q^{92} + 32 q^{95}+O(q^{100}) \) 24 * q - 24 * q^2 + 24 * q^4 - 24 * q^8 + 24 * q^16 + 32 * q^23 - 16 * q^25 - 24 * q^32 - 32 * q^46 + 16 * q^50 - 32 * q^53 + 24 * q^64 - 32 * q^85 + 32 * q^92 + 32 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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